Googling around, it looks like popular solutions with using multiple possible criteria for SUMIF involve using SUMPRODUCT or adding multiple SUMIF functions, but this won't work for what I'm needing.

Without getting into messy details, I'm trying to use a formula that determines if something is "Off Spec". There are multiple categories why something is off spec, and a given record could be off spec for more than one reason. So, trying to use the solutions above would simply count one record twice.

One solution to the issue is having an extra (hidden) column after each entry that does something like =IF(OR(D1 ="Corrosive", E1 >70), "Off", "On"), and then doing a SUMIF off this hidden column. However, I was wondering if there was a way to do the same search from the SUMIF area instead of having to add an extra column of information.